Universality of arm exponents: Theorem 3.4

The proof of universality of arm exponents (Theorem3.4) follows exactly the steps of [GM14, Sec. 8]. Arm events will be transferred between isoradial graphs using the same transforma-tions as in the previous sectransforma-tions. As already discussed in Section2.6, these transformations alter primal and dual paths, especially at their endpoints. When applied to arm events, this could considerably reduce the length of the arms. To circumvent such problems and shield the endpoints of the arms from the effect of the star-triangle transformations, we define a variation of the arm events. It roughly consists in “attaching” the endpoints of the arms to a track which is not affected by the transformations. Some notation is necessary.

Fixε >0and a doubly-periodic isoradial graphG∈ G(ε)with grid(s_n)_n∈Zand(t_n)_n∈Z. Recall that the vertices of G that are below and adjacent tot₀ form the base ofG. Also, recall the notationx↔yand writex←→^∗ yfor connections in the dual configuration.

Forn < N andk∈ {1} ∪2N, define the eventAe_k(n, N)as

1. fork= 1: there exist primal verticesx₁∈Λ(n)andy₁<Λ(N), both on the base, such thatx₁↔y₁;

2. fork= 2: there existx₁, x₁^∗∈Λ(n)andy₁, y₁^∗<Λ(N), all on the base, such thatx₁↔y₁ andx^∗₁←→^∗ y₁^∗;

3. fork= 2j>4: Ae_k(n, N)is the event that there existx₁, . . . , x_j ∈Λ(n)andy₁, . . . , y_j <

Λ(N), all on the base, such thatx_i↔y_ifor alliandx_i=x_j for alli,j.

Notice the resemblance betweenAe_k(n, N)andA_k(n, N), where the latter is defined just before the statement of Theorem3.4. In particular, observe that in the third point, the ex-istence ofj disjoint clusters uniting∂Λ(n)to∂Λ(N)indeed induces2jarms of alternating colours in counterclockwise order. Two differences betweenAe_k(n, N)andA_k(n, N)should

3.3. Proofs for16q64 be noted: the fact that in the former arms are forced to have extremities on the base and that the former is defined in terms or graph distance while the latter in terms of Euclidean dis-tance. As readers probably expect, this has only a limited impact on the probability of such events.

For the rest of the section, fix q ∈ [1,4] and writeϕ_G for the unique infinite-volume random-cluster measure onGwith parametersβ= 1andq.

Lemma 3.20. Fixk∈ {1} ∪2N. There existsc >0depending only onε,q,k and the funda-mental domain ofGsuch that

c ϕG[A_k(n, N)]6ϕG[Ae_k(n, N)]6c⁻¹ϕG[A_k(n, N)] (3.25) for allN > nlarge enough.

The above is a standard consequence of what is known in the field as the arm separation lemma (LemmaD.1). The proofs of the separation lemma and Lemma 3.20are both fairly standard but lengthy applications of the RSW theory of Theorem3.1; they are discussed in AppendixD(see also [Man12, Prop. 5.4.2] for a version of these for Bernoulli percolation).

We obtain Theorem3.4in two steps, first for isoradial square lattices, then for doubly-pe-riodic isoradial graphs. The key to the first step is the following proposition.

Proposition 3.21. LetG⁽¹⁾ =G

α,β⁽¹⁾ and G⁽²⁾ =G

α,β⁽²⁾ be two isoradial square lattices in G(ε). Fixk∈ {1} ∪2N. Then

ϕ^ξ

G⁽¹⁾[Ae_k(n, N)] =ϕ^ξ

G⁽²⁾[Ae_k(n, N)], for alln < N.

Proof. Fixk∈ {1} ∪2Nand takeN > n >0andM>N (one should imagineMmuch larger than N). Let Gmix be the symmetric mixed graph of Section 3.2.1formed above the base of a block ofM rows and2M+ 1columns of G⁽²⁾ superposed on an equal block ofG⁽¹⁾, then convexified, and symmetrically in below the base. Construct Gemix in the same way, with the role ofG⁽¹⁾andG⁽²⁾inversed. Recall, from Section3.2.1, the series of star-triangle transformationsΣ^↓that transformsGmixintoGemix.

Writeϕ_Gmix andϕ_Ge

mix

for the random-cluster measures onGmixandGemix, respectively, withβ= 1and free boundary conditions. The eventsAe_k(n, N)are also defined onGmixand Gemix.

Let ω be a configuration onGmix such thatAe_k(n, N) occurs. Then, under the config-urationΣ^↓(ω) on Gemix, Ae_k(n, N) also occurs. Indeed, the verticesx_i andy_i (and x^∗₁ and y₁^∗ when k = 2) are not affected by the star-triangle transformations in Σ^↓ and connec-tions between them are not broken nor created by any star-triangle transformation. Thus, ϕ_Gmix[Ae_k(n, N)]6ϕ_Ge

mix[Ae_k(n, N)]. Since the roles ofGmixandGemix are symmetric, we find ϕ_Gmix[Ae_k(n, N)] =ϕ

Gemix[Ae_k(n, N)] (3.26) Observe that the quantities in (3.26) depend implicitly onM. When takingM → ∞, due to the uniqueness of the infinite-volume random-cluster measures inG⁽¹⁾andG⁽²⁾, we obtain

ϕ_Gmix[Ae_k(n, N)]−−−−−−→

M→∞ ϕG(1)[Ae_k(n, N)] and ϕ_Ge

mix[Ae_k(n, N)]−−−−−−→

M→∞ ϕG⁽²⁾[Ae_k(n, N)].

Thus, (3.26) implies the desired conclusion.

3. Universality of the random-cluster model

Corollary 3.22. LetG=G_α,β be an isoradial square lattice in G(ε) and fixk ∈ {1} ∪2N. Then, there existsc >0depending only onε,qandksuch that,

c ϕG[A_k(n, N)]6ϕ_Z2[A_k(n, N)]6c⁻¹ϕG[A_k(n, N)], for anyn < N.

Proof. FixG_α,βandkas in the statement. The constantsc_ibelow depend onε,qandkonly.

Letβe_k=α−k−β₀+πand writeeβfor the sequence(βe_k)_k∈Z. Due to the choice ofG_α,β, we haveeβ∈[ε, π−ε]^Z. Proposition 3.21and Lemma3.20applied toZ² =G_0,π

2

andG

0,eβ

yield a constantc₁>0such that c₁ϕG

0,eβ[A_k(n, N)]6ϕ_Z2[A_k(n, N)]6c⁻₁¹ϕG

0,eβ[A_k(n, N)]. (3.27) As in the proof of Corollary 3.12,G_α,β

0 is the rotation byβ₀ of the graphG

0,eβ. This does not imply that the arm events have the same probability in both graphs (since they are defined in terms of square annuli). However, PropositionD.2] about arms extension provides a constantc₂>0such that

c₂ϕ_G

α,β0[A_k(n, N)]6ϕ_G

0,eβ[A_k(n, N)]6c⁻₂¹ϕ_G

α,β0[A_k(n, N)]. (3.28) Finally apply Proposition3.21 and Lemma3.20to G_α,β

0 andG_α,β to obtain a constant c₃>0such that

c₃ϕG_α,β

0[A_k(n, N)]6ϕG_α,β[A_k(n, N)]6c₃⁻¹ϕG_α,β

0[A_k(n, N)]. (3.29) Writing (3.27), (3.28) and (3.29) together yields the conclusion withc=c₁c₂c₃.

Theorem3.4is now proved for isoradial square lattices. To conclude, we extend the result to all doubly-periodic isoradial graphs.

Proof of Theorem3.4. Consider a doubly-periodic graphG∈ G(ε)for someε >0, with grid (s_n)_n∈Zand(t_n)_n∈Z. Fixk∈ {1} ∪2N. The constantsc_ibelow depend onε,q,kand the size of the period ofG.

Choose n < N and M > N (one should think of M as much larger than N). Propo-sition3.9(the symmetrized version) provides star-triangle transformations(σ_k)₁6k6K such that, inGe= (σ_K◦ · · · ◦σ₁)(G), the regionΛ(M)has a square lattice structure. Moreover, each σ_kacts betweens−dMands_dM(for some fixedd>1) and betweent_Mandt−M, none of them affecting any rhombus oft₀.

In a slight abuse of notation (since(s_n)_n∈Z,(t_n)_n∈Zis not formally a grid ineG) we define Ae_k(n, N)forGeas forG.

Letωbe a configuration onGsuch thatAe_k(n, N)occurs. Then, the image configuration (σ_K◦ · · · ◦σ₁)(ω)onGeis such thatAe_k(n, N)occurs. Indeed, the transformations do not affect the endpoints of any of the paths definingAe_k(n, N). Thus,

ϕG[Ae_k(n, N)]6ϕ_eG[Ae_k(n, N)].

The transformations may be applied in reverse order to obtain the converse inequality. In conclusion,

ϕG[Ae_k(n, N)] =ϕ_eG[Ae_k(n, N)]. (3.30)

3.4. Proofs forq >4